False strawberry can be quite invasive. There is some in my mom's yard, and I like it. Biodiversity is a good thing. If you must eradicate this plant, spot spray it with 10 or 20% vinegar. Per one gallon of vinegar add 1 ounce of orange oil and a teaspoon or tablespoon of soap. Soap is better than detergent: something like Ivory I think is all right. (I use Dr. Bronner's peppermint soap). If it gets on the St. Augustine, it will kill it as well, so take care. It is best if you apply the vinegar mixture during the hotttest dryest part of the day.